Nevertheless, one can characterize when one-way distinguishability is possible in the single qubit sender case. As proved in [1]} (Theorem 1), the states must take a particularly nice form, in terms of orthogonality on Alice's side versus corresponding orthogonalities on Bob's side. That said, one could argue that the condition from [1]} is perhaps not so operationally simple to apply to easily identify when states are distinguishable, if the set is large for example. Here we show how, in the case of product states, Theorem REF  can be refined for a single qubit sender to yield an entire graph-theoretic set of testable conditions for one-way distinguishability.
